What is Derivative: Definition and 1000 Discussions

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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  1. A

    I What is the logarithm of the derivative operator?

    I found this article which claims to have found the logarithm of derivative and even gives a formula. But I tried to verify the result by exponentiating it and failed. Additionally, folks on Stackexchange pointed out that the limit (6) in the article is found incorrectly (it does not exist)...
  2. abhinavabhatt

    A Second derivative of Heaviside step function

    In QFT by peskin scroeder page 30 the action of Klein Gordon Operator on propagator (∂2+m2)DR(x-y)=∂2θ(x0-y0)... how to compute this ∂2θ(x0-y0)?
  3. J

    Solving a Partial Derivative Problem Step-by-Step

    So I start by isolating v the speed here would be the square root of the partial t derivative divided by the sum of the partial x and y derivatives. the amplitude, phi and the cos portion of the partial derivatives would all cancel out. What I am left with is the sqrt(43.1 / ( 2.5 + 3.7 ) =...
  4. winnie_d_poop

    Prove that the derivative of the position vector equals the velocity vector

    1.)##\dot{\vec{r}}=\dot{x}\hat{i}+\dot{y}\hat{j}+\dot{z}\hat{k}=\dot{r}\hat{r}## since the unit vector is constant 2.) ##\dot{r}\hat{r}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{\sqrt{x^2+y^2+z^2}}\frac{\dot{x}x+\dot{y}y+\dot{z}z}{\sqrt{x^2+y^2+z^2}}##...
  5. SebastianRM

    Relating volumetric dilatation rate to the divergence for a fluid-volume

    in class we derived the following relationship: $$\frac{1}{V}\frac{dV}{dt}= \nabla \cdot \vec{v}$$ This was derived though the analysis of linear deformation for a fluid-volume, where: $$dV = dV_x +dV_y + dV_z$$ I understood the derived relation as: 1/V * (derivative wrt time) = div (velocity)...
  6. E

    Lie derivative of general differential form

    The first two parts I think were fine, I expressed the tensors in coordinate basis and wrote for the first part$$ \begin{align*} \mathcal{L}_X \omega = \mathcal{L}_X(\omega_{\nu} dx^{\nu} ) &= (\mathcal{L}_X \omega_{\nu}) dx^{\nu} + \omega_{\nu} (\mathcal{L}_X dx^{\nu}) \\ &= X^{\sigma}...
  7. Jason-Li

    Engineering Proportional plus derivative controller

    Hello, There is a thread related to this question however it was marked correct but doesn't look correct to me? https://www.physicsforums.com/threads/step-change-in-a-proportional-plus-integral-controller.961180/ I think I have it but it is quite different to other answers I have seen? I...
  8. Tony Hau

    I The derivative of the complex conjugate of the wave function

    It is a rather simple question: In my textbook it writes something like: $$\frac {\partial \Psi} {\partial t}= \frac{i\hbar}{2m}\frac {\partial^2 \Psi} {\partial x^2}- \frac{i}{\hbar}V\Psi$$ $$\frac {\partial \Psi^*} {\partial t}= -\frac{i\hbar}{2m}\frac {\partial^2 \Psi^*} {\partial...
  9. E

    B A few questions about the covariant derivative

    Hey everyone, I was trying to learn in an unrigorous way a bit about making derivatives in the general manifold, but I'm getting confused by a few things. Take a vector field ##V \in \mathfrak{X}(M): M \rightarrow TM##, then in some arbitrary basis ##\{ e_{\mu} \}## of ##\mathfrak{X}(M)## we...
  10. greg_rack

    Exploring the Sign of a 2nd Derivative Function

    Since the index of the root is odd, the domain is going to be ##R##, and I can calculate the second derivative to be: $$y''=\frac{1}{3}\times \frac{e^x(e^x-3)}{3(e^x-1)^{\frac{5}{3}}}$$ Studying the sign of this function, it results positive for ##x<0 \vee x>ln(3)##, so the main function will be...
  11. greg_rack

    B Find Local Max/Min: 2nd Derivative=0

    How do I distinguish between a point of local maxima or minima, when the second derivative in that point is equal to zero?
  12. greg_rack

    Derivative of a Point of Max: Finding a & b Conditions

    The derivative of a point of maximum must be zero, and since $$y'=3ax^2+2bx+2 \rightarrow y'(-1)=3a-2b+2 \rightarrow 3a-2b+2=0$$ we get the first condition for ##a## and ##b##. Now, since we want ##x=-1## to be a local maximum, the derivative of the function must be positive when tending to...
  13. B

    Lifting a survivor into a helicopter with a rope

    We have 2 forces affecting the rope: 1. Gravitational force of the body ##=mg## and 2. Force of air = Force of drag= ##F_{AIR}##. The length of the rope is shortening with the velocity ##v_k##. So to figure out the angle ##\theta## I wrote: ##R##= force of rope ##R_x = F_{AIR}## ##R_y = mg##...
  14. greg_rack

    Issue calculating the derivative of a rational function

    First, I calculated the derivative of $$D(\sqrt{ax})=\frac{a}{2\sqrt{ax}}$$ Then, by applying the due theorems, I calculated the deriv of the whole function as follows: $$ f'(x)=\frac{\frac{a}{2\sqrt{ax}}(\sqrt{ax}-1)-\sqrt{ax}(\frac{a}{2\sqrt{ax}})}{(\sqrt{ax}-1)^2}=...
  15. J

    Covariant Derivative of a Vector

    Apologies in advance if I mess up the LaTeX. If that happens I'll be editing it right away. By starting off with ##\nabla^{'}_{\mu} V^{'\nu}## and applying multiple transformation laws, I arrive at the following expression $$ \frac{\partial x^{\lambda}}{\partial x'^{\mu}} \frac{\partial...
  16. A

    Problem with a directional derivative calculation

    Good day I have a problem regarding the directional derivative (look at the example below) in this example, we try to find the directional derivatives according to the two approaches ( the definition with the limit and the dot product of the vector gradient and the vector direction) in this...
  17. A

    Problem with a directional derivative calculation

    this is the function and this is the solution in which the definition has been used my question is Why we can not use the traditional approach? I mean calculation the partial derivative which equals 0 in our case? And doing the dot product with the vector v (after normalizing it) many...
  18. T

    I Derivative using the chain rule

    I'm coming back to maths (calculus of variations) after a long hiatus, and am a little rusty. I can't remember how to do the following derivative: ## \frac{d}{d\epsilon}\left(\sqrt{1 + (y' + \epsilon g')^2}\right) ## where ##y, g## are functions of ##x## I know I should substitute say ##u = 1...
  19. F

    I Divergence & Curl -- Is multiplication by a partial derivative operator allowed?

    Divergence & curl are written as the dot/cross product of a gradient. If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator. is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
  20. T

    I Treating the Derivative like a Quotient

    Good Morning I have read that it is not justified to split the "numerator" and "denominator" in the symbol for, say, dx/dt However, when I look at Wikipedia's discussion on the Principle of Virtual Work, they do just that. (See picture, below). I was told it is OK in 1D cases, but note the...
  21. J

    I Variation of Ricci scalar wrt derivative of metric

    I understand from the wiki entry on the Einstein-Hilbert action that: $$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$ What is the following? $$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$ Is there a place I could look up such GR expressions on the internet? Thanks
  22. redtree

    I Integrating with the Dirac delta distribution

    Given \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon} \delta^{(2)}(x-y) f(x) dx &= f^{(2)}(y) \end{split} \end{equation} where ##\epsilon > 0## Is the following also true as ##\epsilon \rightarrow 0## \begin{equation} \begin{split} \int_{y-\epsilon}^{y+\epsilon}...
  23. D

    Numerical approximation of the 2nd derivative across a diffuse interface

    Imagine you create a diffuse interface in space and determine which side of the interface you are on by a local scalar value that can be between 0 and 1. We could create a circle, centered in a rectangular ynum-by-xnum grid, with such a diffuse interface with the following MATLAB code: xnum =...
  24. SchroedingersLion

    A Derivative of norm of function w.r.t real-part of function

    Greetings, suppose we have ##h(u)=\frac{1}{2} \left\|Au-b \right\|_{2}^2## with ##A## a complex matrix and ##b,u## complex vectors of suitable dimensions. Write ##u=u_1 + iu_2## with ##u_1## and ##u_2## as the real and imaginary part of ##u##, respectively. Show that ##\frac {\partial h}...
  25. D

    Calculating 2nd Total Derivative of u w.r.t. t

    du/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) So i write the operator as d/dt = (dx/dt)(∂/∂x) + (dy/dt)(∂/∂y) and apply it to du/dt ; in the operator it is the partial derivative that acts on du/dt which involves using the product rule. I am having a problem with the term involving (∂/∂x)...
  26. T

    Determine for which x the derivative exists of: ##f(x)=\ln|\sin(x)|##

    Hi there. I have the following function: $$f(x)=\ln|\sin(x)|$$ I've caculated the derivative to: $$f'(x)=\frac{\cos(x)}{\sin(x)}$$ And the domain of f(x) to: $$(2\pi n, \pi+2\pi n ) \cup (-\pi + 2\pi n, 2\pi n)$$ And the domain of f'(x) to: $$(\pi n, \pi+\pi n )$$ I want to determine for...
  27. T

    Determine for which x the derivative exists for $$f(x)=arcsin(\sqrt x)$$

    Hi there. I have the following function: $$f(x)=arcsin(\sqrt x)$$ I've caculated the derivative to: $$f'(x)=\frac{1}{2\sqrt x\sqrt{ (1-x}}$$ And the domain of f(x) to: $$[0, 1]$$ And the domain of f'(x) to: $$(0, 1)$$ I want to determine for which x the derivative exists but I'm not...
  28. T

    A Accelerated Hubble expansion -- Is the second derivative positive?

    Since distances increase, their first derivative which is velocity (Hubble constant) should be positive if not increasing too. Accelerated expansion needs the velocity to increase. What about the third derivative which is acceleration? An accelerated universe could have third derivative (called...
  29. R

    I Why Does the Partial Derivative of a Sum Cancel Out?

    Why the summation of the following function will be canceled out when we take the partial derivative with respect to the x_i? Notice that x_i is the sub of (i), which is the same lower limit of the summation! Can someone, please explain in details?
  30. M

    Covariant derivative and the Stress-enegery tensor

    My try: $$ \begin{align*} \nabla^a T_{ab} &= \nabla^a \left(\nabla_{a} \phi \nabla_{b} \phi-\frac{C}{2} g_{a b} \nabla_{c} \phi \nabla^{c} \phi\right)\\ &\overset{(1)}{=} \underbrace{(\nabla^a\nabla_{a} \phi)}_{=0} \nabla_{b} \phi + \nabla_{a} \phi (\nabla^a\nabla_{b} \phi)-\frac{C}{2}...
  31. Frigus

    B Why the derivative of area is related to the graph of the function

    the explanation about the question I got from internet is, A very small change in area divided by the dx will give the function of graph so anti-derivative of function of graph should be equal to the area of the function. It also seem quite obvious to me but I am not satisfied by it, It seems to...
  32. Another

    Classical Mechanics: Why can we cancel the derivative of dt from these equations?

    problem in this book : classical mechanics goldstein Why can we cancel the derivative of dt from these equations? e.g. ##\frac{d(x)}{dt} + \frac{b sin\theta}{2} \frac{d(\theta)}{dt} = asin\theta \frac{d(\phi)}{dt}## ## x +\frac{b \theta sin\theta}{2} = a \phi sin\theta ## because I think...
  33. Neothilic

    I What is the nth derivative of this equation?

    How would I find the ##nth## derivative of this? As as derivative of ##-|t|## is ##-\frac{t}{|t|}##. ## \langle X^{n} \rangle = i^{-n}\frac{d^{n}}{dt^n} e^{-|t|} \vert_{t=0} ## This is the characteristic function of the Cauchy Distribution. So for when ##t=0##, ##e^{-|t|}=1##; when ##t<0##...
  34. Leonardo Machado

    I Chebyshev Differentiation Matrix

    Hi everyone. I am studying Chebyshev Polynomials to solve some differential equations. I found in the literature that if you have a function being expanded in Chebyshev polynomials such as $$ u(x)=\sum_n a_n T_n(x), $$ then you can also expand its derivatives as $$ \frac{d^q u}{dx^q}=\sum_n...
  35. M

    Partial derivative of Vxx w.r.t. r in terms of Vxx

    Can anyone please help me to write partial derivative of Vxx w.r.t. r in terms of Vxx as shown in the hand written box at the end.
  36. S

    B Standard version of covariant derivative properties

    [Throughout we're considering the intrinsic version of the covariant derivative. The extrinsic version isn't of any concern.] I'm having trouble reconciling different versions of the properties to be satisfied by the covariant derivative. Essentially ##\nabla## sends ##(p,q)##-tensors to...
  37. LCSphysicist

    Sign of a second partial derivative

    I am not sure how to determine the sign of this derivatives. (a) first we can pass a plane by (1,2) parallel to XZ (y fixed) and see how the curve belongs to the plane will vary with x, but what about the next partial derivative, with respect to y?
  38. S

    B Defining the derivative of a vector field component

    I'm reading 'Core Principles of Special and General Relativity' by Luscombe, specifically the introductory section on problems with defining usual notion of differentiation for tensor fields. I'll quote the relevant part: Since the equation above is a notational mess, here's my attempt to...
  39. T

    I How can you prove the integral without knowing the derivative?

    Hello (A continued best wishes to all, in these challenging times and a repeated 'thank you' for this site.) OK, I have read that Newton figured out that differentiation and integration are opposites of each other. (This is not the core of my question, so if that is wrong, please let it go.)...
  40. E

    Force on a capacitor plate as the derivative of the energy -- is it a fluke?

    Bit of a random question... capacitors can be a bit weird, in that if we connect one up to a source of EMF and do positive external work to separate the plates of the capacitor, the energy of the capacitor decreases (and instead the work you do plus the decrease in capacitor energy goes into the...
  41. K

    I Derivative of a function of another function

    This is really a simple question, but I'm stuck. Suppose we have a function ##\vartheta'(\vartheta) = \vartheta## and that ##\vartheta = \vartheta(\varphi)## and we know what ##\vartheta(\varphi)## is. How should I view ##\frac{\partial \vartheta'}{\partial \varphi}##? Should I set it equal to...
  42. G

    MHB How to solve for x using 2nd derivative?

    Hi, I am trying to find the minimum root (x) of one formula. For that, I took 2nd derivative and got this equation. \[ 2 \times A \times (\frac{T}{x^3})=0 \] Here A,T,x are greater then 0. I don't know how to proceed further, how to solve it for x? Can you please guide me?
  43. N

    B Question about Limits and the Derivative

    Hello. I bought "Calculus Made Easy" by Thompson and it got me thinking about something I wondered about before. This question is a bit hard for me to articulate, but I'll do my best: When we are trying to find the limit as change in x approaches zero of dy/dx, we take smaller and smaller...
  44. S

    B Reconciling basis vector operators with partial derivative operators

    Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process. Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...
  45. T

    B When I'm "finding the derivative" can it be said that I am "differentiating the equation"?

    I’m am on a path of trying to learn calculus which I should have done long ago. I am making some progress. But I would like to know this... I know what a derivative is. Is differentiation the process of finding a derivative? In other words, when I am finding the derivative can it be said...
  46. P

    B Derivative of a constant scalar field at a point

    Wikipedia defines the derivative of a scalar field, at a point, as the cotangent vector of the field at that point. In particular; The gradient is closely related to the derivative, but it is not itself a derivative: the value of the gradient at a point is a tangent vector – a vector at each...
  47. A

    I Derivative of a definite integral

    If $$F(x)=\int_{a}^{b}f(x)dx$$ implies $$F'(x)=\int_{a}^{b}f'(x)dx$$?
  48. leticia beira

    Finding the derivative of this trig function

    Para f (θ) = √3.cos² (θ) + sen (2θ), uma inclinação da reta tangente, uma função em θ = π / 6, é?
  49. SchroedingersLion

    A Partial / Total Derivative, Compositions

    Hello there, I have stumbled across further examples to derivatives of multivariable functions that confuse me. Similar to my other thread: https://www.physicsforums.com/threads/partial-derivative-of-composition.985371/#post-6309196 Suppose we have two functions, ## f: R^2 \rightarrow R...